Comparative Study of Molecular Dynamics, Diffusion, and

ISSN 1990-7478, Biochemistry (Moscow) Supplement Series A: Membrane and Cell Biology, 2008, Vol. 2, No. 1, pp. 73–81. © Pleiades Publishing, Ltd., 2008. Original Russian Text © K.V. Shaitan, M.Yu. Antonov, Ye.V. Tourleigh, O.V. Levtsova, K.B. Tereshkina, I.N. Nikolaev, M.P. Kirpichnikov, 2008, published in Biologicheskie Membrany, 2008, Vol. 25, No. 1, pp. 66–75.

Comparative Study of Molecular Dynamics, Diffusion, and Permeability for Ligands in Biomembranes of Different Lipid Composition K. V. Shaitana, M. Yu. Antonova, b, Ye. V. Tourleigha, O. V. Levtsovaa, K. B. Tereshkinaa, I. N. Nikolaevb, and M. P. Kirpichnikova a Lomonosov

Moscow State University, Moscow, 119991 Russia e-mail: [email protected], [email protected], [email protected], b Ammosov Yakut State University, ul. Belinskogo 58, Yakutsk, 677000 Russia e-mail: [email protected] Received August 21, 2007

Abstract—A comparative study of several model lipid bilayers of different composition, which included analysis of kinetic parameters of model lipid bilayers and permeability of bilayer membranes for small molecules, has been carried out. The conformity of results of numeric experiments to experimental data (structure of membrane lipid bilayers, lateral diffusion coefficients, and relative permeability of biomembranes for ligands) is discussed in the framework of a standard molecular dynamics protocol. DOI: 10.1134/S199074780801011X

Structural and kinetic properties of biological membranes play a crucial role in cell-mediated processes [1]. In this context, simulation of biomembrane structures by the steered molecular dynamics (MD) method presents considerable interest in view of rapid progress in molecular and membrane technologies [2] and unique opportunities for detailing and visualization of molecular processes in complexly built structures on the basis of advanced numeric experiment protocols [3–5]. The latter includes microscopic mass transfer imaging in strongly anisotropic and complexly structured media, diffusion at membrane/water interface, formation and relaxation of non-equilibrium heterophasic systems, etc. [6, 7].

simulations of membrane lipid bilayers [12] notably reduce the experimental time, but fail to provide reliable kinetic information even when sophisticated heavy-atom lipid bilayer models are used [13, 14]. Of course, nearly all specific simulated structures can be calibrated in such a way that some calculated parameters are brought in full conformity with experimental data, but versatility of MD protocols and possibility to extend them to other objects leave doubt. At the same time, MD simulation of hydrated lipid bilayers of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylcholine (POPC) culminated in the development of a novel efficient MD protocol. This procedure is based on all-atom Amber 1999 force field [15] and is designed for the study of major structural and kinetic parameters of biomembranes. The main feature of the novel approach is to bring the system to a state characterized by equilibrium distribution of fluctuations of major macroparameters, such as temperature, volume, pressure, etc. [16, 17]; its practical utility consists in broadening the range of objects analyzed by the method in question and bringing experimental values of lateral pressure to conformity with previously obtained data [18].

The use of MD simulation in membrane studies [8, 9] is often coupled with considerable difficulties. As molecular models of membrane structures contain no less than 104 atoms, calculation of 100-ns trajectories in all-atom force fields is tedious and time-consuming. Moreover, characteristic times of passive ion transport are usually measured on a microsecond scale. Therefore, a search for novel efficient numeric protocols for obtaining explicit information in a reasonably short time is a currently central task. Several approaches to the solution of this problem exist, but a simple twophase solvation model for fast assessment of miscellaneous effects of hydrophobicity factors on structural interfacial changes in biomolecules [10, 11] seems to be more preferable. However, this approach has one serious disadvantage, viz., it does not take into account the contribution of Coulomb interactions between membrane components and ligands. Coarse-grained

This study is an overview of structural regularities and dynamic behavior of lipid membranes whose lipid compositions are similar to those of higher organisms in the example of three most common membrane lipids and their bilayers with special reference to permeability of biomembranes for low-molecular endogenous ligands. Physical mechanisms of these processes are still poorly understood despite the large body of evidence on permeability of biomembranes for low73

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Table 1. Values of lateral pressure for Systems I–III System

Mean specific area, Å2

Lateral pressure, bar

I II III

78.0 ± 5.5 65.4 ± 1.5 66.8 ± 3.7

–242 –300 –265

Table 2. Calculated values of specific area, bilayer thickness, and isothermal compression coefficients System

Mean membrane thickness, Å

χT, Pa–1

I II III

34.33 ± 0.39 35.54 ± 0.25 36.06 ± 0.32

1.7 × 10–10 1.4 × 10–10 2.1 × 10–10

molecular solutes [19]. In this study, the steered MD (SMD) simulation was used as a method of choice in the analysis of permeabilities of biomembranes for ligands [16, 17, 20, 21]. The procedure consists in applying external force to the ligand and steering the system through a definite evolutional scenario, i.e., monitoring of transmembrane transport even in relatively short trajectories and estimation of parameters which are characteristic for ligand translocation. EXPERIMENTAL The following models of bilayer lipid membranes of different lipid composition were used: 1-palmitoyl-2oleoyl-sn-glycero-3-phosphatidylcholine (POPC), 1,2dipalmitoyl-sn-glycero-3-phosphatidylcholine (DPPC) and 1,3-(1-stearoyl-2-palmitoyl-sn-glycero-3-phosphatidyl)glycerol (cardiolipin, CL). Their trajectories were calculated using a package of molecular dynamics programs (PUMA software) [22, 23]. Classical equations were solved with the help of Verlet algorithm with the Amber 1999 potential field [15]. The calculations were carried out under periodic boundary conditions for three hydrated lipid bilayers: (i) 8 CL molecules, 16 POPC molecules, and 16 DPPC molecules (System I); (ii) 30 POPC molecules and 30 DPPC molecules (System II) and (iii) 64 POPC molecules (System III). The original structures of the lipid bilayers corresponded to the perpendicular position of the principal axis relative to the plane of the membrane. To achieve this, prior to experiments lipid molecules were turned round the long axis to a random angle. For each lipid molecule, there were 34–43 solvent molecules; in the majority of lipids full hydration is reached only when no less than 27 water molecules per one lipid is used [24]. In the initial configuration, water molecules were separated from extreme atoms by a distance no less than 2.3 Å. The original design of the experimental systems afforded full conformity of their

specific areas to calculated values [25–30], e.g., 62– 68 Å2 for POPC, 59–62 Å2 for DPPC, and 100–120 Å2 for CL. In bilayers formed from different lipids, specific areas were calculated as mean areas of participating lipids relative to their concentration. Partial charges and force constants for lipid molecules were calculated as described previously [16]. The negative charge of CL molecules (–2) was compensated by adding Na+ ions to water. The valence bonds and valence angles in water molecules were not fixed (TIP3P model). The van der Waals interactions were analyzed using a special smoothed (switching) function. The value of the Coulomb potential was multiplied by a special shielding function as described in [16]. The value of the cut-off radius for Coulomb interactions varied from 16 to 20 Å depending on membrane type. The value of the dielectric constant was taken equal to unity; the numerical integration step was 1 fs. The calculations were performed under periodic boundary conditions at constant temperature and under constant pressure (NPT ensemble). Barostating was performed in a Berendsen altitude chamber; the relaxation time (100 ps) was the same in all directions. To ensure conformity of the specific area of the lipid bilayer to experimental values [25–30] and to take into consideration the contribution of surface tension of the lipid bilayer, the lateral components of pressure were taken negative [31]. Constant temperature (300 K) was maintained with the help of a collisional medium (collisional thermostat [23]). Mean collision frequency of virtual particles was 10 ps–1; average particles mass was 1 a.m.u. RESULTS AND DISCUSSION In the first series of our experiments, we studied lipid bilayer relaxation (200–500 ps, 500 K) and selected a working part in the MD trajectory no less than 2 ns long. In studies of dynamics, negative lateral pressure was applied to the test systems in order to maintain mean specific area close to initial level (Table 1). Figures 1a–1c show probability densities of volume fluctuations of a calculation cell for the systems considered. As can be seen, the probability density of these fluctuations had the shape of a Gaussian distribution. In terms of Einstein thermodynamic fluctuation theory, the probability density of equilibrium fluctuations of volume p(∆V) is set by the Gaussian distribution: 2

p ( ∆V ) = A V e

( ∆V ) – -----------------2 2 〈 ∆V 〉

.

The figures in broken brackets 〈 〉 represent the derivation of the mean, AV is the normalization factor, 〈∆V2〉 = kbTVχT is the dispersion and χT is the isothermal compression coefficient for the given system. Hence, the value of 〈∆V2〉 was obtained after calculation of χT (Table 2). Previous studies showed that χT for

BIOCHEMISTRY (MOSCOW) SUPPLEMENT SERIES A: MEMBRANE AND CELL BIOLOGY

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COMPARATIVE STUDY OF MOLECULAR DYNAMICS, DIFFUSION p × 103 (‡) 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 9.55 9.60 9.65 9.70 9.75 9.80 V × 10 –4, Å3 S, Å2 81

(d)

80 79 78 77 76 1600 2000 2400 2800 3200 3600 t, ps Z, Å 36.0

(g)

35.5 35.0 34.5 34.0 33.5 33.0 1600 2000 2400 2800 3200 3600 t, ps

p × 103 (b) 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 1.280 1.284 1.288 1.292 1.296 V × 10 –5, Å3 2 S, Å 66.2 (e) 66.0 65.8 65.6 65.4 65.2 65.0 64.8 64.6 1600 1800 2000 2200 t, ps Z, Å 36.6 (h) 36.4 36.2 36.0 35.8 35.6 35.4 35.2 35.0 34.8 1700 1500 1900 2100 t, ps

p × 103 1.2

75

(c)

1.0 0.8 0.6 0.4 0.2 0 1.60 S, Å2 69.0 68.5 68.0 67.5 67.0 66.5 66.0 65.5

Z, Å 37.2 37.0 36.8 36.6 36.4 36.2 36.0 35.8 35.6 35.4 35.2

1.62 1.63 V × 10 –5, Å3

1.61

(f)

3200 3600 4000 4400 4800 t, ps (i)

3200 3600 4000 4400 4800 t, ps

Fig. 1. Parameters of the calculation cell. (a–c) Probability density of calculation cell volume and Gaussian approximation of the experimental curve. (d–f) Specific area fluctuations per lipid molecule. (g–i) Bilayer thickness fluctuations for Systems I–III, respectively.

lipid membranes lies in the range from 1 × 10–10 to 6 × 10–10 Pa–1 [32]. The experimental values of χT also lie within a reasonable range. Membrane thickness (Figs. 1g–1i) was defined as a distance between phosphorus atoms in adjacent monolayers. Their mean values are listed in Table 2. Figure 2 shows the mean-square deviation and linear approximation of displacement of POPC molecules for different experimental systems. The lateral diffusion coefficient of the lipids Dxy is defined as a linear approximation coefficient for the dependence: 〈 ( x ( t + τ ) – x ( t ) ) + ( y ( t + τ ) – y ( t ) ) 〉 = 4D xy t. 2

2

Here, the square deviation of the mass center of a lipid in the plane of the lipid bilayer is put in broken brack-

ets. The averaging was performed individually for each lipid type: 0 < t < T – τ ps, 0 < τ < 300 ps, where T is the length of the working part of the trajectory. The calculated values of Dxy (Table 3) were close to those for quasi-elastic neutron scattering on DPPC (1 × 10–7 cm2/s, [33]) and dioleoylphosphatidylcholine (2 × 10–7 cm2/s, [34]) bilayers. The corresponding values obtained after pulse-modulated NMR of POPC bilayers at 298 and 303 K were 2.0 × 10–7 and 2.5 × 10–7 cm2/s, respectively [35]. Comparison of results obtained for times lesser than 1 ns to neutron scattering data is more correct, since at higher (1 ns) values of this parameter diffusion induces more profound changes in the membrane structure (10 Å) [36], while diffusion coefficients measured by fluorescent methods are 2–


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SHAITAN et al. 〈(x(t) – x(0))2 + (y(t) = y(0))2〉, Å2 3.0 (‡) 2.5

〈(x(t) – x(0))2 + (y(t) = y(0))2〉, Å2 3.0 (b) 2.5

〈(x(t) – x(0))2 + (y(t) = y(0))2〉, Å2 3.0 (c) 2.5

2.0

2.0

2.0

1.5

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0

50 100 150 200 250 300 0 t, ps

50 100 150 200 250 300 0 t, ps

50 100 150 200 250 300 t, ps

Fig. 2. Mean area of POPC geometric center translocation in the plane of the lipid bilayer and its linear approximation for Systems I (a), II (b) and III (c).

Electron density, e/nm3 (‡) 1 400 3 2 300

Electron density, e/nm3 (b) 400 1 300

4 8

200 100 5

6

3

2 4

200 100

9

7

0 –40 –30 –20 –10 0

5

Electron density, e/nm3 1 (c) 400 3 300 2 4 8 200

6

8

100

9

0 10 20 30 40 –40 –30 –20 –10 0 Z, Å

7

5

9

6 7

0 10 20 30 40 –40 –30 –20 –10 0 10 20 30 40 50 Z, Å Z, Å

Fig. 3. Distribution of electron density in Systems I (a), II (b) and III (c). Curve 1, integral system; 2, water; 3, lipid; 4, lipid heads; 5, nitrogen atoms; 6, phosphorus atoms; 7, HC=CH-groups; 8, CH2 groups of alkyl chains; 9, terminal CH3-group of alkyl chains.

3 times as low [37]. Direct comparison is hardly efficient under these conditions, since lipid translocation can be impeded by the label (e.g., rhodamine). After introduction of a double cis-bond (POPC) or an increase in the molecular size (POPC and CL), the lateral mobility of the lipid bilayer descends, weakly but steadily, in the following order: DPPC POPC CL. At the same time, low sensitivity of the lateral difTable 3. Calculated values of lateral diffusion coefficients System I I I II II III

Lipid POPC DPPC CL POPC DPPC POPC

Dxy , cm2/s

Experimental value, cm2/s

(2.2 ± 0.3) × 10–7 – (2.2 ± 0.5) × 10–7 – –7 (2.0 ± 0.3) × 10 – (2.4 ± 0.6) × 10–7 – –7 (2.6 ± 0.7) × 10 1 × 10–7* [33] –7 (2.6 ± 0.5) × 10 (2.0–2.5) × 10–7 [35]

* For monolipid bilayers.

fusion coefficient to the structure of the lipid membrane suggests that the mechanism of lateral diffusion is confined to gradual small-scale displacement of individual components of the lipid membrane and does not involve the whole lipid molecule. Matching of other critical parameters of model lipid bilayers also takes place. This concerns, in particular, distribution of averaged electron and atomic group densities along the normal to the membrane. Similar dependences were established for, e.g., DPPC [38], dioleoylphosphatidylcholine [39] and POPC [28]. The distribution of electron density in membranes of different lipid composition is shown in Fig. 3. In another series of our experiments, we investigated radial atomic distribution g(r) in the plane of the lipid membrane. The function g(r) determines the probability of localization of a certain group of atoms at a definite distance from other atoms projected onto the plane of the membrane. The number of atoms dN in the circular layer with the square dS and thickness dr, which is separated by a distance r from the central atom


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COMPARATIVE STUDY OF MOLECULAR DYNAMICS, DIFFUSION g(r) 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0

g(r) 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

(‡) 2 1

2

4

6

8 10 12 14 16 r, Å

1

g(r) 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

(b) 2

2

0

4

6

8

10 12 14 16 r, Å

0

77 (c) 1 2

4

2

6

8

10 12 14 16 r, Å

Fig. 4. Radial distribution functions of nitrogen and phosphorus atoms in the plane of the membrane for Systems I (a), II (b) and III (c); 1, P–P; 2, N–N.

–SCH

4

0.3

6

–SCH

(‡) 1 8

3

0.2 0.1

0.3

7

2

3

4

0.0

0.0

–0.1

–0.1 0 2 4 6 8 10 12 14 16 18 Number of carbon atom

0.2

1

0.1

(c)

0.3

0.2 5

–SCH

(b)

1

0.1 2

2

0.0 –0.1

0 2 4 6 8 10 12 14 16 18 Number of carbon atom

0 2 4 6 8 10 12 14 16 18 Number of carbon atom

Fig. 5. Profiles of order parameters for C–H-bonds in the alkyl chains of Systems I (a), II (b), and III (c). 1, Palmitoyl chains of POPC; 2, oleoyl chains of POPC; 3 and 4, palmitoyl chains of DPPC; 5 and 6, palmitoyl chains of CL; 7 and 8, stearoyl chains of CL.

(where N is the total number of atoms of the given type), is correlated with g(r) as: dS N dN = Ng ( r ) ------ = ---- g ( r )2πrdr. S S The radial distribution of phosphorus and nitrogen atoms in the plane of the membrane is shown in Fig. 4; the averaging was performed for the both lipid layers. The radial distribution patterns of nitrogen atoms testify to close-spaced localization of nitrogen atoms around the lipid heads of the membrane lipid bilayer, especially if such membranes contain different types of lipids. The thickness of the hydrophilic lipid bilayer is such that weakly charged, mutually repulsive nitrogen atoms are localized at different depths in the membrane lipid bilayer. Actually, heavily charged phosphorus atoms cannot be separated from one another by such a short distance. The magnitude of the first maximum diminishes with the increase in the number of bilayer lipid types as a result of which the orderliness of the lipid bilayer is impaired even if the distance between the atoms is small. In our study, the shapes of the experimental curves testify to the lack of long-range packing of membrane lipid heads (Fig. 4).

The values of the order parameter SCH (Fig. 5), which can be established by NMR spectroscopy of deuterated lipids, were calculated from MD data. Averaged values of this parameter can also be calculated from IR spectroscopic data [6]. The order parameter for the C−H bonds in the alkyl fragments of the phospholipids was determined by the formula: 1 2 S CH = --- 〈 3 cos θ i – 1〉 , 2 where θi is the angle between the C–H bond at the ith carbon atom in the alkyl chain of the phospholipid and the normal to the membrane, while the figures in broken brackets designate time averaging. The maximum value of this parameter (when all bonds are parallel to the normal to the membrane) is equal to unity, the minimum value (when all bonds lie in the plane of the membrane) is –0.5. The calculated values of –SCH for the oleoyl chain manifest themselves as a characteristic depression near the double bond. Dissipative characteristics of lipid membranes and diffusion of small molecules. Formamide, ammonia, water, oxygen, glycerol, ethanediol, ethanol, butyric acid, and urea were studied as candidate ligands


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SHAITAN et al. η, ÒP 6

η, ÒP 6

(‡)

(b)

UREA 5

5 GRL

4 3

4 3

O2

AMM BUTA

2

2

WAT

FMD ETL

1 0 1.5

WAT

AMM O 2 2.0

1

ETD 2.5

3.0

3.5 Ref, Å

0 1.9

2.0

2.1

2.2

2.3 Ref, Å

Fig. 6. Dependence of microviscosity on effective radius Ref in the POPC–water system. Total Fext = 10 kcal/mol Å–1; (a) 6 kcal/mol Å–1; (b) 2 kcal/mol Å–1. Designations: WAT, water; AMM, ammonia; O2, molecular oxygen; FMD, formamide; ETD, ethanediol; ETL, ethanol; UREA, urea; GRL, glycerol; BUTA, butyric acid.

(penetrants) using steered molecular dynamics simulation as a method of choice [20, 21]. Additional potential was superimposed on the system in order to stimulate its translocation along selected degrees of freedom. A constant potential (2 and 6 kcal/mol Å–1) applied to aqueous solutions of tested compounds was oriented along the normal in the direction of membrane surface. In the latter case (6 kcal/mol Å–1), the measurements were performed in triplicate. The force was applied uniformly to all atoms of the ligand. The simulations were run till the first full penetration of ligand molecules into the membrane, but no longer than 2 ns. Displacement of ligand molecules was assessed by drift. Diffusion was not taken into consideration; local friction coefficients were determined as a ratio of applied force to drift velocity: F ext -. γ = -------ν The friction coefficient may conveniently be expressed as medium microviscosity using the Stokes formula or as a diffusion coefficient using the Einstein relation: kbT kbT -. = -----------F = -------6πηr γ The factors responsible for the significant deviation of true values from the Stokes hydrodynamic formula were described in our previous publication [17]. Interestingly, the Stokes ratio was found to be qualitatively more efficient even at the microlevel [40]. As was discussed previously [17], there exists a certain critical level of superimposed force, which favors transmembrane transfer of particles into the membrane in the course of nanoseconds (e.g., 1 kcal/mol Å–1 for

2A particles). The calculated values of effective viscosity descend at increasing force, which testify to nonnewtonian origin of the medium and low non-equilibrium state of the experimental system at flow rates varying from 1 to 10 A/ps. The dependences of effective microviscosities on ligand type and applied potential are shown in Fig. 6. The higher the effective microviscosity of the lipid membrane at the given value of applied force, the lower the permeability of the membrane for ligand molecules. At 2 kcal/mol Å–1, only very small molecules can penetrate into the membrane within a course of several nanoseconds, this time being too short for larger molecules to overcome the membrane barrier (Fig. 6b). Experimentally determined microviscosity values for membrane lipids vary from 30 to 190 cP depending on membrane type [41–43]. The corresponding value for POPC is of the order of 18 cP [44]. However, there is little or no evidence on microviscosities measured along the normal. If ligand transport occurs in the hydrodynamic regime, membrane microviscosity does not depend either on the size or the chemical structure of the ligand molecule. Deviations can be induced by specific interactions between the ligand and the membrane (Fig. 6). A common feature is that the radius of the low-molecular ligand varies directly as the effective microviscosity of the membrane. This conclusion is in agreement with the increase in effective microviscosity, which is inversely correlated with the external force and, as a consequence, with the rate of ligand molecules transfer through the membrane. Special mention should be made of the drastic decrease in the permeability of membranes for urea, which can be attributed to strong dipole-dipole interactions creating local traps for urea molecules.


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79

logP (‡) BUTA

–1

(b)

–1

BUTA

AMM

–2 WAT

–2 WAT –3

–3

–5 GRL –6

ETD

–5 UREA GRL

UREA –5

FMD

–4

FMD ETD

–4

–4

–3

–2 –1 log(D × K/∆x)

–8

–7

–6

–5

–4

–3 –2 log(D × K/∆x)

Fig. 7. Dependence of permeability P on DK/∆x, where D is the diffusion coefficient in the membrane, K is the distribution coefficient for the water–hydrophobic solvent system, and ∆x is the membrane thickness (~40 Å). (a) Olive oil (correlation coefficient, 0.999); (b) hexadecane (correlation coefficient, 0.963). Designations as in Fig. 6.

A comparison of experimental mobility data reflecting the behavior of small molecules in hydrophobic solvents to experimental values of diffusion coefficients demonstrated that membrane permeability is a product of the diffusion coefficient and interfacial (membrane/water) distribution coefficient normalized by membrane thickness. Figure 7 illustrates conformity of the calculated interfacial distribution coefficients to experimentally determined values. As calculation of distribution coefficients is not a task in this particular case, the distribution coefficient of the penetrant was calculated from the corresponding values for water– hexadecane and water–olive oil systems [45]. The data obtained were consistent with the previously reported data on permeability of lipid membranes for ligand molecules. Noteworthy, the DK/∆x values for the olive oil–water system showed a better correlation to the permeability P than those for the hexadecane– water system. In this study, the MD approach based on the use of a standard molecular simulation protocol was employed for a comparison of major structural and kinetic parameters of fully hydrated POPC, DPPC and CL bilayers. The use of this protocol brings the system to local equilibrium and ensures easy distribution and stable characteristics of tested parameters. The thickness of the membrane lipid bilayer, the distribution of atomic groups within the membrane relative to the normal, the radial distribution functions in the plane of the lipid bilayer and the order parameters of lipid chains are all consistent with experimental data. The design of collisional thermostats and Berendsen anisotropic barostats affords compensation for surface tension and natural parameterization errors imposed by the force field. The MD protocol used in this study

enables acquisition of reasonable values of lateral diffusion coefficients and permeability of low-molecular ligands consistent with experimental data. ACKNOWLEDGMENTS The authors thank Dr. Yu.N. Antonenko for helpful discussions. This work has been carried out with the financial support from the Federal Agency for Education, the Federal Agency for Science and Innovations and the Russian Foundation for Basic Research (grant nos. 0604-08136 and 07-04-01169) and support from the U.S. Civilian Research and Development Foundation (grant no. 2803). REFERENCES 1. Gennis, R.B., Biomembrany: Molekuliarnaya struktura i funktsii (Biomembranes: Molecular Structure and Function, New York: Springer-Verlag, 1989), Moscow: Mir, 1997. 2. Baker, R.W., Membrane Technology and Applications, N. Y.: McGraw-Hill, 1999. 3. Karplus, M. and McCammon, J., Molecular Dynamics Simulations of Biomolecules, Nature Struct. Biol., 2002, vol. 9, pp. 646–652. 4. Frenkel, D. and Smit, B., Understanding Molecular Simulation: From Algorithms to Applications, 2nd ed., San Diego: Acad. Press, 2002. 5. Shaitan, K.V., Tourleigh, Ye.V., Golik, D.N., Tereshkina, K.B., Levtsova, O.V., Fedik, I.V., Shaytan, A.K., Li, A., and Kirpichnikov, M.P., Nonequilibrium Molecular Dynamics of Nanostructures Including Biological Ones, Khim. Fizika (Rus.), 2006, vol. 25, pp. 31–48.


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